Abstracts for Singularities in Materials, October 25-29, 2004
University of Minnesota
University of Minnesota
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Mathematics of Materials and Macromolecules: Multiple Scales, Disorder, and Singularities, September 2004 - June 2005

Abstracts:

IMA Workshop:

Singularities in Materials

October 25-29, 2004

Photo Gallery     Materials from Talks

Mark J. Ablowitz (Department of Applied Mathematics, University of Colorado) mark.ablowitz@colorado.edu

Wave Collapse in Nonlocal Nonlinear Schrödinger Equations

Wave collapse occurs in nonlinear media whose governing equations have quadratic nonlinearities. Examples include water waves and nonlinear optics. Although these two physical problem are very different, the equations that govern their dynamics are similar. The equations, which couple the first harmonic to the mean terms in a quasi-monochromatic amplitude perturbation expansion are nonlocal nonlinear Schrodinger systems. They are sometimes referred to as Benney-Roskes or Davey-Stewartson type. The two dimensional ground state solution, is found to play an important role in the wave collapse mechanism.

Stan Alama (Department of Mathematics and Statistics, McMaster University) alama@univmail.cis.mcmaster.ca

Vortices and Pinning Effects for the Ginzburg-Landau Model in Multiply Connected Domains
Slides:   pdf

We consider the two-dimensional Ginzburg-Landau model with magnetic field, for a superconductor with multiply connected cross-section. We study energy minimizers in the London limit as the Ginzburg-Landau parameter tends to infinity, to determine the number and asymptotic location of vortices. We show that the holes act as pinning sites, acquiring nonzero winding for bounded fields and attracting all vortices away from the interior for fields up to a critical value. At the critical level the pinning effect breaks down, and vortices appear in the interior of the superconductor at locations which we identify explicitly, in terms of the solutions of an elliptic boundary value problem. The method involves sharp upper and lower energy estimates, and a careful analysis of the limiting problem which captures the interaction between the vortices and the holes.

Suman Anand (Polymeric and Soft Material Division, National Physical Laboratory) sanand@mail.nplindia.ernet.in

Investigation of the Spectral Properties at Phase Singularity (poster)

Joint work with M.N.Kamalasanan.

Recently optical vortices have attracted considerable attentions because of the phase singularity and the characteristic intensity distribution. In particular, the dark region is very useful to trap and guide atoms. Several methods of generating the optical vortices have been reported so far.

We have demonstrated a simple generation of optical vortices produced by computer-generated holograms near the focus of a converging spatially fully coherent polychromatic wave using a simple experimental technique. The anomalous behavior of the spectra at and in the vicinity of phase singularity has also been studied. It is found that the spectrum of the wave on the hologram, consisting of a single spectral profile splits into two peaks at the on-axis point and shows red shift and blue shift around the phase singularity. These spectacular spectral changes also take place in the vicinity of dark rings of Airy pattern formed with spatially coherent, polychromatic light diffracted at a circular aperture. In another experiment, it is also demonstrated that spectral degree of coherence of partially coherent light may posses isolated points at which its phase is singular, and that in the neighborhood of these points the phase may possess a vortex structure. Here, also we found that at the point of phase singularity the spectrum splits into two peaks.

Fabrice Bethuel (Laboratoire Jacques-Louis Lions, ENPC-CERMA ) bethuel@ann.jussieu.fr

Slow Motion Collisions and Phase Vortex Interaction in the Two-Dimensional Parabolic Ginzburg-Landau Dynamics

The talk is based on a joint work with D.Smets and G. Orlandi, in which we describe a natural framework for the vortex dynamics the parabolic complexGinzburg-landau equation on the plane. This general setting does not rely on any assumption of well-preparedness and has the advantage to be valid even after possible collision times. We analyze carefully collisions leading to annihilation, and identify a new phenomenon, the phase-vortex interaction, related to persistency of low frequency oscillations, and leading to an unexpected drift in the motion of vortices.

I will also discuss open issues, as the splitting of multiple degree vortices,, collisions without annihilation, ...

Rustum Choksi (Department of Mathematics, Simon Fraser University) choksi@cs.sfu.ca

Phase Separation in Block Copolymer Systems: Paradigms for Multiscale Phase Separation
Slides:   pdf

Systems of block copolymers are paradigms for phase separation on several scales and display a wide range of geometric structures. Their remarkable ability for self-assembly (in the melt phase) into various ordered structures aids in the construction of many so called "designer materials"; Indeed, these geometrical structures are key to the many properties that make diblock copolymers of great technological interest.

This talk will be devoted to certain macroscopic models for phase separation in diblock copolymer melts and homopolymer blends, their derivation, and the mathematical questions that arise. In particular, within the context of these models I will discuss both the periodicity and scale of these patterns, and their geometry. The latter appears to be closely related to the periodic isoperimetric problem which, in three dimensions, remains an open problem in classical geometry.

Parts of the talk will cover joint work with X. Ren (Utah State University), joint work with G. Alberti (University of Pisa) and F. Otto (University of Bonn), and work in progress with P. Sternberg (Indiana University).

Alan Dorsey (Department of Physics, University of Florida) dorsey@phys.ufl.edu

Electronic "Liquid Crystals"

There is growing experimental evidence that electrons confined to two dimensions (in a semiconductor heterostructure, for instance) at low temperatures and high magnetic fields can display a plethora of partially ordered phases which have the same symmetries as classical liquid crystal phases, such as nematics and smectics. I will review the experimental evidence for these novel quantum phases of matter, and discuss their phenomenology, with an emphasis on the key role of topological defects (dislocations) partially restoring broken symmetries.

Slides from a related talk - December 2003 colloquium at NYU:   html    pdf    ps    ppt

Andre Geim (Centre for Mesoscience and Nanotechnology, University of Manchester) geim@man.ac.uk

Sub-Atomic Movements of Domain Walls in Crystal-Lattice Potential
Slides:   pdf

The discrete nature of the crystal lattice bears on virtually every material property but it is only when the size of condensed-matter objects ­- e.g., dislocations, vortices in superconductors, domain walls in magnetic materials - becomes comparable to the lattice period, that the discreteness reveals itself explicitly. The associated phenomena are usually described in terms of the Peierls (?atomic washboard?) potential, which was first introduced for the case of dislocations at the dawn of the condensed-matter era. Since then, it has been invoked in many situations to explain certain features in bulk properties of materials but never observed directly. We have succeeded for the first time to monitor experimentally how a single domain wall moves through individual peaks and troughs of the atomic landscape. The wall becomes trapped between adjacent crystalline planes, which results in its propagation by distinct jumps matching the periodicity of the Peierls potential (Nature 426, 812, 2003). As a domain wall moves from one Peierls valley to another, it becomes unexpectedly flexible at Peierls ridges, which we attribute to atomic-size kinks propagating along a wall in the transient bistable position. The physics of topological defects at this true atomic scale is badly understood and seems to require a theory beyond the existing models.

Tiziana Giorgi (Department of Mathematical Sciences, New Mexico State University) tgiorgi@nmsu.edu

Superconductors Surrounded by Normal Materials (poster)

We consider a generalized Ginzburg-Landau energy functional modeling a superconductor surrounded by a material in the normal state. In this model the order parameter is defined in the whole space. We derive existence of a global minimizer in weighted Sobolev spaces for both square integrable and constant applied magnetic fields. We then prove boundedness and classical elliptic estimates for the oder parameter, in order to study the loss of superconductivity for high applied magnetic fields. In two-dimensions for the general case and in three-dimensions for the case of constant permeability, we show the existence of an upper critical field above which the only finite energy weak solutions are the normal states. For the three-dimensional case, we show that as the applied field tends to infinity finite energy weak solutions tend to the normal state.

Boaz Ilan (Department of Applied Mathematics, University of Colorado at Boulder) boaz@colorado.edu

Singularity Formation in Nonlinear Schrodinger Equations with Fourth-Order Dispersion
Slides:   pdf

Similarly to the classical NLS, NLS equations with fourth-order dispersion can admit singularity formation. Such equations arise in the studies of optical-beam propagation in fiber arrays and in the numerical analysis of finite-difference discretizations of the NLS. We extend the classical-NLS global-existence theory to the NLS with isotropic and anisotropic fourth-order dispersion. Using the results on the critical exponent, it is shown how fourth-order dispersion can assist in the physical realization of spatio-temporal "light bullets" in a pure Kerr medium. These are recent works with Gadi Fibich, George Papanicolaou, Steve Schochet, and Shimshon Bar-Ad.

Sookyung Joo (IMA Postdoc) sjoo at ima.umn.edu http://www.ima.umn.edu/~sjoo/

The Phase Transitions Between the Chiral Nematic and Smectic Liquid Crystals (poster)

We study the Chen-Lubensky model to investigate the phase transition between chiral nematic and smectic liquid crystals. First, we prove the existence of the minimizers in an admissible set where the order parameter vanishes on the boundary. The splay, twist, and bend Frank constants are considered to diverge near N* -- C* phase transition based on physical observation, while only twist and bend constants can be assumed to diverge near N* -- A* transition. Then we describe the transition temperatures for both smectic A side and smectic C side when a domain is a considerably large liquid crystal region confined in two plates.

Robert V. Kohn (Department of Mathematics, Courant Institute) kohn@courant.nyu.edu

A Deterministic-Control-Based Approach to Interface Motion
Slides:  pdf    Paper:   pdf

Interface motion is central to many application areas, including materials science and computer vision. Motion by curvature is a basic example, in which the normal velocity of the interface is equal to its curvature. The level-set approach, now 15 years old, represents the evolving interface as the zero-level-set of an evolving PDE. This viewpoint has been extremely successful for both simulation and analysis.

I'll discuss joint work with Sylvia Serfaty, which develops a new perspective on the level-set approach to motion by curvature and related interface motion laws. We show, loosely speaking, that the level-set PDE is the value function of a deterministic two-person game. More precisely, we give a family of discrete-time, two-person games whose value functions converge in the continuous-time limit to the solution of the motion-by-curvature PDE. This result is unexpected, because the value function of a deterministic control problem is normally a first-order Hamilton-Jacobi equation, while the level-set formulation of motion by curvature is a second-order parabolic equation.

Richard Kollar (IMA Postdoc) kollar@ima.umn.edu http://www.ima.umn.edu/~kollar/

Linear Stability of Vortices in Bose-Einstein Condensates (poster)

The existence of localized vortices in Bose-Einstein condensates was experminetally and analytically confirmed in the previous years. Although there is a large literature on their linear stability, the rigorous and complete approach to this problem is absent. We study the simplest case of a single localized vortex trapped in an harmonic trap in the two-dimensional approximation. We use the Evans function method which proved to be a robust and reliable technique for studying nonlinear eigenvalue problems. We confirm that singly-quantized vortices are linearly stable and that the linear stability of multi-quantized vortices depends on the diluteness of a condensate, with alternating intervals of stability and instability. Moreover, we propose a significant reduction of the numerical cost of the algorithm by replacing the traditional winding number calculation by using the information on the Krein signature of possible unstable eigenvalues. (This is a joint work with Robert L. Pego.)

Matthias Kurzke (IMA Postdoc) kurzke@ima.umn.edu http://www.ima.umn.edu/~kurzke/

Boundary Vortices in Thin Magnetic Films (poster)

In a certain thin-film limit, the micromagnetic functional reduces to a competition between a Dirichlet term and a boundary term penalizing nontangential magnetization. We show that in the limit where the boundary term is strongly penalized, minimizing magnetizations develop two half-vortices at the boundary. Quite similar to results for Ginzburg-Landau interior vortices, we can show that the singular part of the energy depends only on the number of vortices, while their interaction is encoded in the next order term, a renormalized energy like that of Bethuel-Brezis-Helein for Ginzburg-Landau.

Robert P. Lipton (Department of Mathematics, Louisiana State University) lipton@math.lsu.edu

Optimal Lower Bounds on the Electric-Field Concentration in Random Dielectric Media (poster)

The study of failure initiation in heterogeneous random dielectrics requires one to assess the effect of the local electric field concentrations generated by macroscopic potential gradients. Macroscopic quantities sensitive to the local field behavior include higher order moments of the electric field inside the composite. In this context the effective dielectric constant can be viewed as a weighted first moment of the electric field.

The results presented here concern two-phase dielectric composites and provide optimal lower bounds on all higher moments of the electric field. The bounds depend upon the statistics of the random media gathered from image analysis. The moments can be of infinite order. An optimal lower bound on the essential supremum of the magnitude of the electric field is presented that depends explicitly on the two-point statistics of the random composite.

Alexander A. Nepomnyashchy (Department of Mathematics, Technion - Israel Institute of Technology) nepom@techunix.technion.ac.il

Feedback Control of Singularities in Models of Directional Solidification

Joint work with V. Gubareva (Technion), A.A. Golovin (Northwestern University), and V. Panfilov (University of Nevada, Reno)

The nonlinear evolution of the long-wave morphological instability in a directional solidification front characterized by a small segregation coefficient in some cases does not saturate, leading to the formation of "deep cells". In the framework of the Sivashinsky equation, this effect manifests itself through a subcritical instability of a plane solidification front and the development of a singularity in a finite time ("blow-up"). We investigate the possibility of the suppression of the subcritical instability and the blow-up behavior by means of a feedback adjustment of the pulling speed or temperature gradient. We show that the feedback control eliminates the appearance of deep cells and leads to the formation of localized structures at the crystal-melt interface.

Also, we investigate the possibility of the elimination of singularities in the nonlinear development of a subcritical pulsatile instability which may appear in the rapid solidification. In the framework of the subcritical complex Ginzburg-Landau equation, we find that the feedback control may lead to the formation of solitary waves and multipulse waves.

Peter Palffy-Muhoray (Liquid Crystal Institute, Kent State University) mpalffy@cpip.kent.edu

Photonic Band Gap Aspects of Liquid Crystals

Liquid crystals with modulated ground states are periodic dielectric structures, and hence they are self-assembled photonic band gap materials. Fluorescent emission is suppressed in the stop band, but is enhanced at the band edges, and, above a pump threshold, distributed feedback lasing occurs. We interpret experimental observations of reflectance, fluorescence and lasing in terms of the density of states, and give a detailed description of this for helical cholesteric liquid crystals. Lasing is due to the singularity in the density of states at the band edges.

Xingbin Pan (Department of Mathematics, East China Normal University) amaxbpan@zju.edu.cn

Nucleation of Smectics

Similarity of Ginzburg-Landau model of superconductivity and Landau-deGennes model for liquid crystals was discovered by P. G. de Gennes, and has been investigated by many physicists and mathematics. In this talk we shall discuss possible similarity in the mathematical theory of nucleation of superconductivity and that of smectics. We shall also discuss some recent progress on the estimates of the critical number Qc3 for liquid crystals, that is analogy of the critical field Hc3 for superconductivity.

Francois Peeters (Departement Fysica, Universiteit Antwerpen (Campus Drie eiken), Universiteitsplein 1, B-2610 Antwerpen) francois.peeters@ua.ac.be http://www.cmt.ua.ac.be/

Vortex Matter in Nanostructured and Hybrid Superconductors
Slides:   html    pdf    ps    ppt

The interplay between superconductivity and the inhomogeneous magnetic field generated by nanostructured ferromagnets leads to new vortex arrangements not found in homogeneous superconductors. We consider two situations: 1) a ferromagnetic disk on top of a thin superconducting film, and 2) a lattice of such ferromagnetic disks separated by a thin oxide layer on top of a thin superconducting film.

For a single ferromagnetic disk magnetized perpendicular to the plane of the superconducting film we found that antivortices are stabilized in shells around a central core of vortices (or a giant vortex) with size/magnetization-controlled `magic numbers.' The transition between the different vortex phases occurs through the creation of a vortex-antivortex pair under the edge of the magnetic disk. In the case of a lattice of ferromagnetic disks, the antivortices form a rich spectrum of lattice states. In the ground state the antivortices are arranged in the so-called matching configurations between the ferromagnetic disks while the vortices are pinned to the ferromagnets. The exact (anti)vortex structure depends on the size, thickness and magnetization of the magnetic dots, periodicity of the ferromagnetic lattice and properties of the superconductor expressed through the effective Ginzburg-Landau parameter kappa*. The experimental implications of our results such as magnetic-field-induced superconductivity will be discussed.

The theoretical analysis is based on a numerical .exact. solution of the phenomenological Ginzburg-Landau equations.

Dmitry Pelinovsky (Department of Mathematics, McMaster University, Canada) dmpeli@math.mcmaster.ca http://dmpeli.math.mcmaster.ca/

Discrete Solitons and Vortices in Nonlinear Schrödinger Lattices Preprint 1:   pdf    Preprint 2:   pdf

We consider the discrete solitons and vortices bifurcating from the anti-continuum limit of the discrete nonlinear Schrodinger (NLS) lattice. The discrete soliton in the anti-continuum limit represents an arbitrary finite superposition of in-phase or anti-phase excited nodes, separated by an arbitrary sequence of empty nodes. The discrete vortices represent a finite set of excited nodes with non-pi phase shifts between the adjacent nodes.

By using stability analysis, we prove that the discrete solitons are all unstable near the anti-continuum limit, except for the solitons, which consist of alternating anti-phase excited nodes. We classify analytically and confirm numerically the number of unstable eigenvalues associated with each family of the discrete solitons.

By using Lyapunov-Schmidt reductions, we study existence and stability of symmetric, super-symmetric and asymmetric discrete vortices in simply-connected discrete contours on the plane.

Peter Philip (IMA Postdoc) philip@ima.umn.edu http://www.ima.umn.edu/~philip/

Optimal Contol of a PDE with Weakly Singular Integral Operator: Tuning Temperature Fields During Crystal Growth (poster)
Slides:   pdf

We consider the optimal control of the temperature gradient during the modeling of diffuse-gray conductive-radiative heat transfer. The problem arises from the aim to optimize crystal growth by the physical vapor transport (PVT) method. The temperature is the solution of a semilinear PDE where radiative heat trasfer between cavity surfaces is modelled by a weakly singular integral operator. Based on a minimum principle for the semilinear equation, we establish the existence of an optimal solution as well as necessary optimality conditions. The theoretical results are illustrated by results of numerical computations.

Daniel Phillips (Department of Mathematics, Purdue University) phillips@math.purdue.edu

Existence of Classical Solutions of the Time--Dependent Ginzburg--Landau Equations for a Bounded Superconducting Domain in a Vacuum

Joint work with Patricia Bauman and Hala Jadallah.

The initial value problem for the Ginzburg--Landau equations modeling currents in a three--dimensional body surrounded by a vacuum is a mixed, elliptic--parabolic system. We prove existence, uniqueness, and regularity results near the body's surface for solutions. Moreover we compare solutions from different gauges.

Yves Pomeau (Department of Mathematics, University of Arizona) yves.pomeau@lps.ens.fr

Finite Time Singularities of Solutions of the Euler Fluid Equations in 3D

Joint work with D. Sciamarella.

The smoothness at all times of the solutions of the equations for inviscid incompressible fluids in 3D with finite energy remains a major unanswered question in applied mathematics. Long ago Leray suggested to look at the equations for self-similar singularities for viscous fluids, by assuming a point singularity in 3D. This way of approaching the question has been taken very rarely over the years. I will show, in the case of 3D Euler, that such a point singularity is very unlikely because of the conservation of energy and circulation, that puts very severe constraints on the possible singularities. Following this line of thought, one discovers that a singularity is possible along a line at a given time. Furthermore, a pure self similar collapse is also unlikely because it brings in a stationary flow with constant positive divergence in the collapsing frame. If nothing counteracts this positive divergence, anything is pulled to infinity in the long run and there is no non trivial solution of the self similar equations. Therefore one needs to have some sort of instability that brings features at smaller and smaller scales as one gets closer and closer to the singularity time. I'll sketch a possible scenario for such a quasi-self similar collapse in axissymetric geometry.

Jacob Rubinstein (Department of Mathematics, Indiana University) jrubinst@indiana.edu http://www.math.indiana.edu/people/profile.phtml?id=jrubinst

The Least Action Principle for Waves

The least action principle of Maupertuis, Lagrange, Hamilton and others lies at the foundation of classical mechanics. Given the initial and terminal points of a system of particles, the orbit of the particles is determined by minimizing the action. I shall describe a generalization of this principle that applies to waves. The principle will be derived first for smooth solutions of the Schroedinger equation, and then generalized to other wave equations and to singular solutions.

Peter Takac (Fachbereich Mathematik, Universitaet Rostock ) takac@hades.math.uni-rostock.de

The Time-Dependent Ginzburg-Landau Equations in the Energy Space W1,2 (omega) (poster)

We show the existence, uniqueness and well-posedness of mild solutions to the time-dependent Ginzburg-Landau equations (TDGL) in the natural "energy space" W1,2 (omega). This requirement is motivated by the fact that the Ginzburg-Landau free energy should be finite. If the applied magnetic field and its time derivative are bounded in time (e.g., time-periodic) as functions valued in L2(omega), then so is the solution of the TDGL equations. The solution is strong for all positive times. To obtain our results, we use abstract evolutionary equations with all terms valued in L3/2 (omega). The smoothing action of the heat semigroup is realized in L3/2 (omega).

Sylvia Serfaty (Courant Institute of Mathematical Sciences, New York University) serfaty@courant.nyu.edu

Vortex-Dynamics in the Ginzburg-Landau Heat-Flow

The Ginzburg-Landau equations are a model for superconductivity, in which the crucial singularities are topological singularities of vortex type. We will describe some results obtained with Etienne Sandier on the dynamics of vortices in Ginzburg-Landau, which are obtained by a Gamma-convergence-based method of convergence which gives criteria to prove that solutions to the gradient-flow of energies which Gamma-converge to the solutions of the gradient-flow of the limiting energy. We will also describe some results on vortex-collisions.

Jack Xin (Department of Mathematics, University of Texas - Austin) jxin@math.utexas.edu

The (2+1) sine-Gordon Equation and Dynamics of Localized Pulses
Paper:    pdf

The (2+1) sine-Gordon (SG) equation is derived from a Maxwell-Bloch system to model the dynamics of localized light pulses in two space dimensional cubic nonlinear materials. A class of nonlinear Schroedinger equations with both focusing and defocusing mechanisms appears as underlying asymptotic description for both the propagation and interaction phenomena.The dynamical persistence of pulses is related to the internal oscillations. Alternative asymptotic methods are reviewed to shed more light. Numerical simulations show the robustness of solitary wave like interaction on the plane.

Emmanuel Yomba (Department of Physics, Faculty of Sciences, University of Ngaoundere) eyomba@yahoo.com

New exact Solutions for the Coupled Klein-Gordon-Schoedinger Equations (poster)

The mapping method is used with the aid of symbolic computation system Mathematica for constructing exact solutions to the coupled Klein-Gordon-Schroedinger equations. The solutions obtained in this work include Jacobi elliptic solutions, combined Jacobi elliptic solutions, triangular solutions, soliton solutions and combined soliton solutions.

Wendy Zhang (Department of Mathematics and Statistics, Oakland University) w2zhang@oakland.edu http://www.oakland.edu/~w2zhang

Surface Faceting Induced by Instability and Singularity in Evolution Equations (poster)

We study the effect of anisotropic surface free energy on the evolution of surface morphology where the transport mechanism is surface diffusion. Based on Herringâ^Ā^Ųs model, we formulate an evolution equation, a partial differential equation, in a global coordinate system. The partial differential equation is nonlinear, time-dependent and is of 4th order in space. The anisotropy is classified into three cases, mild, critical and severe anisotropy, according to the stability and singularity of the evolution equation. We show that when the anisotropy is mild, the evolution equation is stable and the surface is smooth. When the anisotropy is critical, surface corners and edges appear in the evolution. When the anisotropy is severe, small surface facets and coarsening occur. The results of numerical computation of surface morphologies are compared to experimental observations.